Applying Boltzmann's definition of entropy

Abstract

Boltzmann defined entropy by the formula where is the volume of phase space occupied by a thermodynamic system in a given state. He postulated that is proportional to the probability of the state, and deduced that a system is in its equilibrium state when entropy is a maximum. This paper attempts to compute directly in a number of interesting and practical cases by adopting a more mathematical viewpoint than is conventional. This allows the student a firmer grasp of the concept of entropy without losing sight of the physical interpretation. By maximizing the entropy, the following results are deduced: the equation of state of an ideal monatomic gas, the exponential decay of atmospheric pressure with height, and a known but not widely appreciated generalization of Trouton's rule governing the equilibrium of a liquid with its vapour.